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Pure Maths 1,2,3 - Coggle Diagram

- - - - Exact values (0°, 30°, 45°, 60°, 90°)
    - - sin²θ + cos²θ = 1
      - tanθ = sinθ / cosθ
  - - - Parallel: m₁ = m₂
      - Perpendicular: m₁·m₂ = –1
    - - distance = √[(x₂–x₁)² + (y₂–y₁)²]
      - midpoint = ((x₁+x₂)/2, (y₁+y₂)/2)
    - - y = mx + c
      - ax + by + c = 0
  - - - Sₙ = ½ n (a + l) (l = last term)
      - Sₙ = ½ n [2a + (n–1)d]
      - uₙ = a + (n–1)d
    - - Sₙ = a(1–rⁿ)/(1–r) (r≠1)
      - S∞ = a/(1–r) (|r|<1)
      - uₙ = arⁿ⁻¹
    - - (a+b)ⁿ = Σ nCr aⁿ⁻ʳ bʳ
      - (1+x)ⁿ = 1 + nx + n(n–1)/2! x² + … (|x|<1 if n not positive integer)
  - - - Translations
      - Stretches / reflections
      - absolute value
  - - - Maximum / minimum (first & second derivative tests)
      - Points of inflection
  - - - Discriminant (Δ = b² – 4ac)
        
        Δ = 0 : equal roots (tangent)
        
        Δ < 0 : no real roots
        
        Δ > 0 : two distinct real roots
      - x = [–b ± √(b² – 4ac)] / 2a
- - - - Concepts
        
        Modulus (absolute value) function
        
        Piecewise definition, graph (reflection of x‑axis part)
        
        Polynomial division
        
        Remainder theorem
        
        Factor theorem
        
        Partial fractions
        
        Linear distinct factors
        
        Repeated linear factors
        
        Quadratic factor (no real roots)
        
        Binomial expansion
        
        Convergence condition: |x| < 1 (for rational n)
      - Formula
        
        Remainder: f(x) ÷ (ax–b) → remainder f(b/a)
        
        Factor: if f(b/a)=0 → (ax–b) is a factor
        
        Partial fractions (3 standard forms)
        
        (1+x)ⁿ = 1 + nx + n(n–1)/2! x² + … (rational n, |x|<1)
    - - Formula
        
        logₐ(xy) = logₐx + logₐy
        
        logₐ(x/y) = logₐx – logₐy
        
        logₐ(xᵏ) = k logₐx
        
        logₐb = log꜀b / log꜀a (change of base)
        
        ln e = 1, e^(ln x) = x, ln(eˣ) = x
        
        aˣ = e^(x ln a)
    - - Formula
        
        sec x = 1/cos x, cosec x = 1/sin x, cot x = 1/tan x = cos/sin
        
        sec²x = 1 + tan²x
        
        cosec²x = 1 + cot²x
        
        sin(A±B) = sinA cosB ± cosA sinB
        
        cos(A±B) = cosA cosB ∓ sinA sinB
        
        tan(A±B) = (tanA ± tanB) / (1 ∓ tanA tanB)
        
        sin2A = 2 sinA cosA
        
        cos2A = cos²A – sin²A = 2cos²A – 1 = 1 – 2sin²A
        
        tan2A = 2tanA / (1 – tan²A)
        
        R = √(a²+b²), tan α = b/a (careful with quadrant)
    - - Formula
        
        d/dx(e^(ax+b)) = a·e^(ax+b)
        
        d/dx(ln(ax+b)) = a/(ax+b)
        
        d/dx(sin x)=cos x, d/dx(cos x)=–sin x, d/dx(tan x)=sec²x
        
        d/dx(sec x)=sec x tan x
        
        d/dx(cosec x)=–cosec x cot x
        
        d/dx(cot x)=–cosec²x
        
        Product: d(uv)/dx = u dv/dx + v du/dx
        
        Quotient: d(u/v)/dx = (v du/dx – u dv/dx)/v²
        
        P3 only: d/dx(tan⁻¹x) = 1/(1+x²)
      - Common Mistakes
        
        Implicit: forgetting to multiply by dy/dx when differentiating y‑terms
        
        Chain rule: missing the inner derivative for ln(ax+b) or e^(ax+b)
        
        Confusing tan⁻¹x (arctan) with (tan x)⁻¹ = cot x
    - - Formula
        
        ∫e^(ax+b) dx = (1/a)e^(ax+b) + C
        
        ∫1/(ax+b) dx = (1/a)ln|ax+b| + C
        
        ∫sin(ax+b) dx = –(1/a)cos(ax+b) + C
        
        ∫cos(ax+b) dx = (1/a)sin(ax+b) + C
        
        ∫sec²(ax+b) dx = (1/a)tan(ax+b) + C
        
        P3: ∫1/(x²+a²) dx = (1/a)tan⁻¹(x/a) + C
        
        P3: ∫u dv/dx dx = uv – ∫v du/dx dx (integration by parts)
        
        Trapezium rule: ∫ₐᵇ f(x) dx ≈ h/2 [y₀ + yₙ + 2(y₁+…+yₙ₋₁)], h = (b–a)/n
      - Common Mistakes
        
        Forgetting +C in indefinite integrals
        
        P3: ∫1/(x²+a²) → forgetting the factor (1/a)
        
        P3: Integration by parts – choosing wrong u and dv (use LIATE order)
        
        P3: Substitution – forgetting to change dx and limits (for definite)
        
        Trapezium rule: mis‑counting intervals (h = (b–a)/n)
        
        Integration of partial fractions: sign errors in partial fraction decomposition
    - - Formula
        
        |a| = √(a₁²+a₂²+a₃²)
        
        a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cosθ
        
        Perpendicular: a·b = 0
        
        Line equation: r = a + tb (t is parameter)
        
        Distance between two points = |AB| = |b – a|
    - - Concepts
        
        First‑order differential equations (dy/dx = f(x,y))
        
        Variables separable (dy/dx = f(x)g(y))
        
        General solution (contains +C)
        
        Particular solution (using initial/boundary condition)
      - Formula
        
        Separate: ∫ 1/g(y) dy = ∫ f(x) dx
    - - Concepts
        
        Definition: z = x + iy (i² = –1)
        
        Real part (Re) and imaginary part (Im)
        
        Complex conjugate (z̄ = x – iy)
        
        Arithmetic (addition, subtraction, multiplication, division)
        
        Modulus |z| = √(x²+y²)
        
        Argument θ = arg(z) (principal value: –π < θ ≤ π)
        
        Polar (mod‑arg) form: z = r(cosθ + i sinθ) = r e^(iθ)
      - Formula
        
        i² = –1
        
        Division: multiply numerator & denominator by conjugate of denominator
        
        z·z̄ = |z|²
        
        |z₁z₂| = |z₁||z₂|, arg(z₁z₂) = arg(z₁) + arg(z₂)
        
        |z₁/z₂| = |z₁|/|z₂|, arg(z₁/z₂) = arg(z₁) – arg(z₂)
        
        Euler form: z = r(cosθ + i sinθ)
    - - Convergence criteria
        
        Iterative methods (xₙ₊₁ = g(xₙ))
        
        Locating roots by sign change (interval bisection / decimal search)